Particle Method for the McKean-Vlasov equation with common noise

Th\'eophile Le Gall (CEREMADE)

REVIEW 3 major objections 5 minor 30 references

Particle Method for the McKean-Vlasov equation with common noise

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A common-noise mean-field SDE can be simulated by a two-stage Euler-plus-particle scheme whose strong error is $C(h^{1/2\wedge\rho}+E_N)$.

desk verdict Clean Euler rate for common-noise McKean-Vlasov, but the headline particle sup_t rate has a proof gap in Section 4; likely fixable, needs revision before I'd trust the main theorem. read the letter →

arxiv 2412.17418 v1 pith:U4ZA2GZF submitted 2024-12-23 math.NA cs.NAmath.PR

Théophile Le Gall (CEREMADE) This is my paper · ORCID

classification math.NAcs.NAmath.PR MSC 60H1065C3065C35

keywords McKean–VlasovequationcommonnoiseparticlemethodEulerschemepropagationofchaosWassersteindistanceempiricalmeasuremeanfieldgames

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that the McKean–Vlasov equation with common noise can be simulated by freezing the coefficients on an Euler grid and then replacing the conditional law in the coefficients by the empirical measure of $N$ particles, and that the combined strong error is bounded by a time-discretisation term $h^{1/2\wedge\rho}$ plus a particle term $E_N$ that decays like $N^{-1/(2p)}$, $N^{-1/d}$, or a log-corrected version depending on the dimension. The point is that adding a common noise — a Brownian driver shared by all particles and affecting the coefficients through the conditional law — does not destroy the clean error separation known for the standard McKean–Vlasov case. If the bound is right, users can choose the number of particles and the time step to balance the two error sources, and the method applies to coefficients, such as $|x|$ or a square-root-like diffusion, that schemes requiring differentiability of the diffusion cannot handle.

What carries the argument

The auxiliary non-interacting particle system (4.1): $N$ conditionally i.i.d. copies $\bar{Y}^i$ of the continuous Euler scheme, all driven by the same common noise $W^0$. These copies play the role of the ideal particle cloud: by Lemma 4.1 their empirical measure $\nu^N$ has the same conditional law as the Euler scheme's law, so classical empirical-measure estimates apply directly; the interacting particle system is shown to stay within a constant multiple of $\nu^N$'s Wasserstein error, and the Euler scheme is shown to stay within $h^{1/2\wedge\rho}$ of the true solution.

What would settle it

Construct a common-noise coefficient whose conditional law has a $(p+\varepsilon)$-moment that grows without bound along some common-noise path, run the particle method (1.6)–(1.7) with fixed time step and increasing $N$, and measure $\sup_{t\in[0,T]}W_p(\bar{\mu}^N_t,\bar{\mu}_t)$; if this supremum decays slower than $N^{-1/(2p)}$ (or the appropriate dimension-dependent rate), the uniform-in-time step in Theorem 1.2(ii) fails.

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Extended reading notes

Core claim

The central result is Theorem 1.3: for $p\in[2,\infty)$, under Lipschitz continuity in the state and measure arguments and $\rho$-Hölder continuity in time, the first particle path $\bar{X}^{1,N}$ driven by the same Brownian motions as the true solution $X$ satisfies $\|\sup_{t\in[0,T]}|\bar{X}^{1,N}_t-X_t|\|_p \le C(h^{1/2\wedge\rho}+E_N)$. The Euler half of the rate is Proposition 3.1, and the particle half is Theorem 1.2(ii): at the grid times, the $p$-Wasserstein distance between the particle empirical measure $\bar{\mu}^N_{t_m}$ and the conditional law $\bar{\mu}_{t_m}$ decays at the classical empirical-measure rate, with an exponent that depends on whether $p$ is larger, equal, or smaller than $d/2$. The proof route is to compare the interacting particles with $N$ conditionally i.i.d. copies of the continuous Euler scheme, transfer the empirical-measure error to the interacting system through a Gronwall-type estimate, and then add the Euler error.

Load-bearing premise

The load-bearing premise is that the conditional empirical measure $\nu^N$ approximates the Euler scheme's conditional law uniformly over the whole time interval at the same rate as at a single time, with an error constant that stays integrable in the common-noise randomness.

Editorial extensions

If this is right

For a fixed computational budget, the error bound shows how to balance the time step and the particle number: once $h^{1/2\wedge\rho}$ is comparable to $E_N$, refining time without adding particles no longer reduces the leading error.
The method requires no differentiability of the diffusion coefficients, so non-smooth maps such as $x\mapsto|x|$ or a square-root-like diffusion are admissible under the stated assumptions.
Because the Euler discretisation keeps the measure argument symbolic, other spatial discretisations, such as quantization, can replace the empirical measure without redoing the time-error analysis.
In dimensions $d<2p$ the particle error decays only as $N^{-1/d}$, so the particle count dominates the total error once the time step is small; in higher dimensions the $N^{-1/(2p)}$ rate applies.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct check of the uniformity step in Theorem 1.2(ii) would compute $\sup_{0\le t\le T} E_1[W_p^p(\bar{\mu}_t(\omega^0),\nu^N_t(\omega^0))]$ for a non-Gaussian example and compare the empirical slope with $E_N^p$; if the slope is worse, the uniform-in-time step, not the single-time rate, is the fragile part.
Because the paper's scheme freezes the measure argument at grid points, replacing the empirical measure by a quantization grid, as already done for the no-common-noise version, should inherit the same $h^{1/2\wedge\rho}$ time error; this is an extension the author points toward but does not prove.
The result is a strong error in the shared-Brownian-motion coupling; it does not by itself give an $L^p$ bound on the distance between marginal laws, so density estimation or weak error would need a separate argument.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a genuine extension of the Liu Euler-particle analysis to common noise, with a clean Euler rate and a particle rate that is probably correct but whose written proof has a real gap. Theorem 1.3 as written does not follow.

What is new: an Euler scheme that keeps the measure argument, Lipschitz coefficients only, p>=2, and rate h^{min(1/2,rho)}. That covers cases like sigma(x)=|x| which the Milstein-type scheme in [4] excludes. Proposition 3.1 is solid: standard Gronwall/BDG with conditional measures, no hidden circularity. The comparison with [4] is accurate, and the numerical examples are illustrative but honest. The conditional-law setup follows Carmona-Delarue correctly.

The soft spot is in Section 4. Theorem 1.2(ii) is supposed to give a sup_t rate. The proof applies Fournier-Guillin pointwise in t and then jumps to sup_t with an integrable constant; the uniform-in-t step and the integrability of the constant are not proved. That looks fixable, but it is not in the paper. More seriously, Theorem 1.3 claims sup_t ||Xbar^{1,N}-Y^1||_p <= C sup_t W_p(mubar^N_t, mubar_t) from (4.4). But (4.4) is fed by g(t) built from W_p(mubar_r, nu^N_r), where nu^N is the empirical measure of the non-interacting Euler copies, not from the interacting mubar^N. The right-hand side should involve sup_t W_p(mubar_t, nu^N_t). Then (4.8) claims a continuous-time sup_t bound for mubar^N vs mubar from Theorem 1.2, but Theorem 1.2 states only a grid sup. So the main sup_t bound is not established as written. This is a load-bearing gap in the proof of the headline particle rate, though the ingredients are very close: (4.4) plus a genuine sup_t bound on W_p(mubar, nu^N) would close Theorem 1.3. I would not call the result false; I would call the proof incomplete.

Bottom line: worth a serious referee. The Euler result alone justifies the paper. The particle theorem needs a patch before publication. A careful referee should ask for a rewritten Section 4 with a real sup_t Fournier-Guillin estimate and the corrected empirical measure in (4.4)/(4.8). Anyone using this should treat the particle rate as conditional on that fix.

Referee Report

3 major / 5 minor

Summary. This paper proposes and analyzes a numerical scheme for McKean-Vlasov stochastic differential equations with common noise. The scheme first discretizes time by an Euler scheme for the conditional-law SDE (1.4)-(1.5) and then approximates the conditional law by the empirical measure of N interacting particles (1.6)-(1.7). The main results are Proposition 3.1, giving an Euler error O(h^{1/2 ∧ ρ}) under Lipschitz and Hölder assumptions; Theorem 1.2, giving rates for the Wasserstein distance between the particle empirical measure and the conditional law; and Theorem 1.3, combining these into a strong error bound of order h^{1/2 ∧ ρ} + E_N for the particle method. Two numerical examples, a conditional Ornstein-Uhlenbeck process and an interbank market model, are provided along with simulation code.

Significance. If valid, Theorem 1.3 gives a clean separation of the total strong error into a time-discretization error and an empirical-measure (particle) error for McKean-Vlasov equations with common noise, extending the framework of Liu [24] to the conditional-law setting. The rates are derived from external benchmarks (Carmona-Delarue conditional-law results, Fournier-Guillin empirical measure bounds, and standard BDG/Gronwall inequalities) rather than fitted to the target quantity, and the paper carefully contrasts its assumptions with the Milstein-type scheme of Biswas et al. [4]. The proof strategy is natural and the claimed rates are plausible. However, as detailed below, two steps in the proof of the particle error are not justified as written, so the central claims require revision rather than immediate acceptance.

major comments (3)

[Section 4, proof of Theorem 1.3 (after Eq. (4.7))] The proof invokes inequality (4.4) to claim that ||sup_t |Xbar^{1,N}_t - Ybar^1_t|||_p is bounded by C sup_t ||Wp(μbar^N_t, μbar_t)||_p, but inequality (4.4) in the proof of Theorem 1.2 bounds this quantity by a constant times g(t), where g is defined through sup_r Wp(μbar_r, ν^N_r) with ν^N the empirical measure of the non-interacting particles Y^i, not through the interacting empirical measure μbar^N. Consequently the displayed chain leading to (4.8) and the final bound (1.9) is not justified as written. The proof can be repaired by using the rate for sup_t ||Wp(μbar_t, ν^N_t)||_p obtained in the proof of Theorem 1.2(ii) and by deriving a continuous-time version of (4.5), but as it stands the central estimate of Theorem 1.3 does not follow.
[Section 4, proof of Theorem 1.2(ii) (application of Lemma 4.3)] The paper asserts that for every ω0 ∈ Ω0, sup_{0≤t≤T} E1[W_p^p(μbar_t(ω0), ν^N_t(ω0))] ≤ C(...) with C independent of N. Lemma 4.3 provides a bound whose constant is C M_q^{p/q}(μbar_t(ω0)) with q=p+ε, so the constant in the supremum-in-t bound is A(ω0)=C sup_t M_q^{p/q}(μbar_t(ω0)), which depends on ω0. Passing to the Lp(P0) rate for sup_t ||Wp(μbar_t, ν^N_t)||_p by the dominated convergence theorem requires integrability of A (or a similar domination). This is not proved in the text; it can be derived from Lemma 3.2 because A(ω0) ≤ (E1[sup_t |Ybar^1_t(ω0,·)|^q])^{p/q}, but the missing argument means the sup_t rate in Theorem 1.2(ii) is not fully established as written.
[Section 4, proof of Theorem 1.2 (final display) and Theorem 1.3 (Eq. (4.8))] Theorem 1.2 is stated for sup_{m∈{1,...,M}} at grid times, yet the proof's final display and the estimate (4.8) in Theorem 1.3 use continuous-time suprema such as sup_{t∈[0,T]} ||Wp(μbar_t, ν^N_t)||_p and sup_{t∈[0,T]} ||Wp(μbar^N_t, μbar_t)||_p. The inequality (4.5) controls only the grid-time supremum, and the continuous-time extension is asserted without proof. Since Theorem 1.3 depends on this extension, it should be stated and proved explicitly.

minor comments (5)

[Section 3, proof of Proposition 3.1 (Eqs. (3.3)-(3.6))] The same symbol s is used for the continuous time and for the grid point floor(s); adopting distinct notation (e.g., \underline{s}) would remove ambiguity.
[Section 1.3, Theorem 1.2] The third-case condition 'p ∈ (0, d/2)' is inconsistent with the standing assumption p ≥ 2; the intended condition is d > 2p.
[Section 5.2, Interbank market model] The text states that the volatility with respect to the local shock is σρ and with respect to the global shock is σ√(1-ρ²), but equation (5.5) has σ√(1-ρ²) dW^i_t and σρ dW^0_t; the verbal description is reversed and should be corrected.
[References] Reference [20] is missing publication details (publisher and year).
[Section 5, figures] The figure captions report empirical slopes but do not discuss their relation to the theoretical rates h^{1/2∧ρ} and E_N; a brief comment would help the reader assess the simulations.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence rates derive from external empirical-measure and stochastic-analysis benchmarks, with no fitted parameter or self-citation carrying the argument.

full rationale

The derivation chain is self-contained against external benchmarks. The Euler rate in Proposition 3.1 is proved from Assumptions 1-3 via Minkowski's inequality, the Burkholder-Davis-Gundy inequality, and Lemma 2.5; no target rate is inserted as an input. The particle rate in Theorem 1.2(ii) is derived by first comparing the interacting particle empirical measure with the non-interacting Euler scheme and then invoking Lemma 4.3, the external Fournier-Guillin empirical-measure bound, for particles that are conditionally i.i.d. given the common noise. That external result has assumptions that do not include the theorem being proved. Theorem 1.3 combines the Euler error of Proposition 3.1 with the propagated particle error; again, the error term E_N is the Fournier-Guillin rate inherited through the comparison argument, not a fitted or normalizing quantity. The skeptical concerns about the proof, such as the substitution of \bar\mu^N for \nu^N in the invocation of (4.4) and the use of the continuous-time estimate (4.8) although Theorem 1.2 is stated at grid times, are potential correctness gaps in the written argument, not circular reductions. There is also no load-bearing self-citation: references to [24], [9], [15], and [20] are independent sources, and the paper does not invoke a prior uniqueness theorem from the same authors to force its construction. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard strong-solution theory for common-noise McKean-Vlasov equations, the conditional-law framework of Carmona-Delarue, the Fournier-Guillin empirical measure bounds, and the Lipschitz/Holder assumptions. No free parameters are fitted to data; no new entities are introduced.

assumptions (7)

domain assumption X0 has finite p-th moment, and finite (p+epsilon)-th moment for the rate results.
Assumption 1; controls the moment growth in (1.8) and the Fournier-Guillin rate.
domain assumption b, sigma and sigma0 are Lipschitz in (x, mu) uniformly in t with respect to |x-y| + W_p(mu, nu).
Assumption 2; used throughout Sections 3-4 for the contraction estimates.
domain assumption b, sigma and sigma0 are rho-Holder in time with factor (1 + |x| + W_p(mu, delta_0)).
Assumption 3; yields the h^rho term in the Euler error and the final h^{min(1/2, rho)} rate.
standard math Strong existence and uniqueness of (1.1) under Assumptions 1-2, quoted from [9, Proposition 2.8].
Used in Lemma 3.4 and Proposition 3.1 to compare X and the Euler scheme.
standard math Burkholder-Davis-Gundy, Minkowski, and the 'a la Gronwall' lemma (Lemma 2.5).
Standard stochastic analysis tools invoked throughout the proofs.
standard math Fournier-Guillin empirical measure rate (Lemma 4.3) applied to conditionally i.i.d. particles.
External theorem used to obtain the N-dependence; the uniform-in-t strengthening is not proven.
standard math Conditional law framework of Carmona-Delarue [9] for L1(X) given the common noise.
Provides existence and continuity of conditional laws used in the measure argument of the coefficients.

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Pith. "Pith review of Particle Method for the McKean-Vlasov equation with common noise." pith.science (2026). https://pith.science/paper/U4ZA2GZF

@misc{pith2026241217418,
  author       = {Pith},
  title        = {Pith review of: Particle Method for the McKean-Vlasov equation with common noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U4ZA2GZF}},
  note         = {Machine review of arXiv:2412.17418}
}

read the original abstract

This paper studies the numerical simulation of the solution to the McKean-Vlasov equation with common noise. We begin by discretizing the solution in time using the Euler scheme, followed by spatial discretization through the particle method, inspired by the propagation of chaos property. Assuming H{\"o}lder continuity in time, as well as Lipschitz continuity in the state and measure arguments of the coefficient functions, we establish the convergence rate of the Euler scheme and the particle method. These results extend those for the standard McKean-Vlasov equation without common noise. Finally, we present two simulation examples : a modified conditional Ornstein Uhlenbeck process with common noise and an interbank market model.

Figures

Figures reproduced from arXiv: 2412.17418 by the authors.

**Figure 4.** 10 paths of particles (X i,N t )t∈[0,T] 16 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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An explicit Milstein-type scheme for interacting particle systems and McKean- Vlasov SDEs with common noise and non-differentiable drift coefficients

S. Biswas et al. “An explicit Milstein-type scheme for interacting particle systems and McKean- Vlasov SDEs with common noise and non-differentiable drift coefficients”. In: Ann. Appl. Probab. 34.2 (2024), pp. 2326–2363. issn: 1050-5164

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work page 2024

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Rate of convergence of a particle method to the solution of the McKean-Vlasov equation

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